equivalent conditions for uniform integrability

Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $S$ be a bounded subset of $L^{1}(\Omega,\mathcal{F},\mu)$. That is, $\int|f|\,d\mu$ is bounded over $f\in S$. Then, the following are equivalent.

1. 1.

For every $\epsilon>0$ there is a $\delta>0$ so that

 $\int_{A}|f|\,d\mu<\epsilon$

for all $f\in S$ and $\mu(A)<\delta$.

2. 2.

For every $\epsilon>0$ there is a $K>0$ satisfying

 $\int_{|f|>K}|f|\,d\mu<\epsilon$

for all $f\in S$.

3. 3.

There is a measurable function $\Phi\colon\mathbb{R}\rightarrow[0,\infty)$ such that $\Phi(x)/|x|\rightarrow\infty$ as $|x|\rightarrow\infty$ and

 $\int\Phi(f)\,d\mu$

is bounded over all $f\in S$. Moreover, the function $\Phi$ can always be chosen to be symmetric and convex.

So, for bounded subsets of $L^{1}$, either of the above properties can be used to define uniform integrability. Conversely, when the measure space is finite, then conditions (2) and (3) are easily shown to imply that $S$ is bounded in $L^{1}$.

To show the equivalence of these statements, let us suppose that $\int|f|\,d\mu for $f\in S$.

(1) implies (2)

For $\epsilon>0$, property (1) gives a $\delta>0$ so that $\int_{A}|f|\,d\mu<\epsilon$ whenever $f\in S$ and $\mu(A)<\delta$. Choosing $K>L/\delta$, Markov’s inequality gives

 $\mu(|f|>K)\leq K^{-1}\int|f|\,d\mu\leq L/K<\delta$

and, therefore, $\int_{|f|>K}|f|\,d\mu<\epsilon$.

(2) implies (3)

For each $n=1,2,\ldots$, property (2) gives a $K_{n}$ satisfying

 $\int(|f|-K_{n})_{+}\,d\mu\leq\int_{|f|>K_{n}}|f|\,d\mu\leq 2^{-n}.$

Without loss of generality, the $K_{n}$ can be chosen to be increasing to infinity, so we can define $\Phi(x)=\sum_{n}(|x|-K_{n})_{+}$. Then,

 $\int\Phi(f)\,d\mu=\sum_{n}\int(|f|-K_{n})_{+}\,d\mu\leq\sum_{n}2^{-n}=1.$

(3) implies (1)

First, suppose that $\int\Phi(f)\,d\mu for $f\in S$. For $\epsilon>0$, the condition that $\Phi(x)/|x|\rightarrow\infty$ as $|x|\rightarrow\infty$ gives a $K>0$ such that $\Phi(x)/|x|\geq 2M/\epsilon$ whenever $|x|>K$. Setting $\delta=\epsilon/2K$,

 $\begin{split}\displaystyle\int_{A}|f|\,d\mu&\displaystyle\leq\int_{|f|>K}|f|\,% d\mu+K\mu(A)\\ &\displaystyle<(\epsilon/2M)\int_{|f|>K}\Phi(f)\,d\mu+K\delta\\ &\displaystyle<\epsilon/2+\epsilon/2=\epsilon.\end{split}$

whenever $\mu(A)<\delta$ and $f\in S$.

Title equivalent conditions for uniform integrability EquivalentConditionsForUniformIntegrability 2013-03-22 18:40:17 2013-03-22 18:40:17 gel (22282) gel (22282) 7 gel (22282) Theorem msc 28A20